What are the islets of Langerhans? Why are these areas more richly supplied with blood vessels?

Put simply, those are the cells in the pancreas that make insulin. The islets of Langerhans contain beta cells, which are sensitive to how much sugar or glucose is washing around in the bloodstream. They tailor how much insulin they make. Insulin is a protein that comes out of the cells, and more is produced the higher the glucose levels. The insulin comes out of the cells and enters the bloodstream through the rich supply of blood vessels. It then goes around the bloodstream telling the cells in the rest of the body to turn on a special transporter, which draws glucose inside the cell rather like a vacuum cleaner. Once it's in the cell, the glucose is then turned into other things like fats and a bigger molecule called glycogen.